Control of magnetic bearing-supported rotors

ABSTRACT

Rotor systems are provided comprising rotor, magnetic bearings and a magnetic bearing controller. The controller is one which varies in accordance with linear parameters and is preferably responsive to linear matrix inequalities. High speed, low loss flywheel systems, especially electromagnetic flywheel batteries are facilitated through such rotor systems.

FIELD OF THE INVENTION

[0001] This invention relates to methods of controlling the performanceof magnetic bearing-supported rotors especially these which are designedto change speeds during operation. High speed rotors are supported bymagnetic bearings which require an electronic control system. Thecontrol systems of this invention can be used to control rotordisplacements and magnetic bearing currents inter alia, at low values.The methods of controlling the magnetic bearing systems of thisinvention employ mathematical relationships deriving from therotor-magnetic bearing system. When a rotor operates over a speed range,parameters in the rotor change with time due to such effects asgyroscopics. The rotor-magnetic bearing systems of the invention employadvanced mathematical modeling and controller development to enable thephysical hardware to operate in the most desirable manner with lowvibration levels and low operating magnetic bearing currents. Thisinvention concerns methods of using a linear parameter varying controlsystem. The present invention also allows the user to implement a smallto moderate size control system on a personal computer or workstationsize computer for the operation of a high speed rotor-magnetic bearingsystem over an operating speed range.

BACKGROUND OF THE INVENTION

[0002] Historically, high speed rotors supported in magnetic bearingshave been employed in applications such as energy storage flywheels,momentum transfer flywheels, pointing devices, and control momentgyroscopes. Typically, rotors in devices such as these are constructedof a relatively massive wheel with high inertia, such as a disk orcylinder, which is attached to a support shaft. The support shaft isdriven by a motor, or motor-generator in the case of energy storageflywheels, and is supported by magnetic bearings. In normal operation,the rotor mass has different spin speeds which produces the energystorage, momentum transfer, pointing and control moment functions. Suchhigh speed, magnetically suspended rotor systems are known per se. Forexample, see U.S. Ser. No. 09/248,520 filed Feb. 2, 1999, incorporatedherein by reference.

[0003] Thus, this invention provides novel rotor-magnetic bearingcontrol, especially linear parameter varying control systems for highspeed rotors. An exemplary application comprising a large inertia energystorage/momentum mechanical flywheel rotor, a motor/generator, a set ofmagnetic bearings to support the flywheel rotor, a support shaftattached to the flywheel rotor, power amplifiers. It is therefore aprimary objective of this invention is to provide improvements in theoperation of high speed rotor-magnetic bearing systems using the linearparameter varying control system. The linear parameter varying controlmethod optimizes rotor operation over the entire speed range of theflywheel plant supported on magnetic bearings. The advantages of theinvention include the reduction of rotor vibration over the operatingspeed range of the rotor compared with previous control methods andminimization of coil currents to minimize rotating power losses in theflywheel rotor operational range.

SUMMARY OF THE INVENTION

[0004] The present invention provides innovative methods of controllingmagnetic bearings, which permits optimum control. Linear parametervarying control methods optimize rotor operation over the entire speedrange of the flywheel plant supported on magnetic bearings. Advantagesof the invention include the reduction of rotor vibration over theoperating speed range of the rotor compared with previous controlmethods and minimization of coil currents to minimize rotating powerlosses in the flywheel rotor operational range.

[0005] Conventional automatic control systems applied to magneticbearing in previous applications assume that the plant is invariant withtime. This means that the control algorithm is formulated based upon theengineering model of the rotor, bearing, actuator, sensor, and othercomponents of the flywheel system, called the plant, that is timeindependent. Some of the control algorithms typically employed for timeindependent control algorithms are proportional-integral-derivative(PID) controls, mu synthesis and H_(∞) controls. The control algorithmis then designed to operate for the specific values of the plant whichare evaluated for a particular speed.

[0006] A number of high speed rotor applications such as high inertiaflywheel rotors inherently have large gyroscopic effects. For example,in energy storage flywheel rotors, the energy stored in the flywheel isproportional to the mass of the flywheel, the square of the rotationalspeed, and the square of the radius of the mass. This means that themost effective energy storage will have most of the mass at the outeredges of the flywheel. In turn, this means that gyroscopic effects arevery significant in energy storage/momentum wheels. Gyroscopic forces inenergy storage/flywheel rotors (or any rotating mechanical component)are proportional to the operating speed of the rotor. These forcescouple the vibration of the rotor along two transverse axes, oftendenoted x and y, which makes these forces difficult to control with themagnetic bearing control algorithm and maintain a centered rotoroperation.

[0007] Other rotor properties will also vary with rotor operationalspeed. The rotor geometry changes due to centrifugal stress and thermalexpansion. As the speed increases, centrifugal stresses induce strainsin the rotor material generating significant increases in the rotordiameter and other dimensions. As the rotor speed increases, heatgeneration will increase and the rotor will run at higher temperatures.Thermal expansion of the rotor material creates additional changes inrotor geometry.

[0008] An energy storage/momentum flywheel must have the capability toaccelerate and decelerate at a certain rate in order to have tocapability to attain the desired energy storage or production and/or thedesired changes in momentum. As the flywheel rotor accelerates ordecelerates, the rotor properties change compared to conventionalconstant speed rotor models.

BRIEF DESCRIPTION OF THE DRAWINGS

[0009]FIG. 1 depicts an exemplary high speed magnetic bearing-supportedrotor useful, e.g. in an energy storage flywheel.

[0010]FIG. 2 shows certain magnetic bearing components.

[0011]FIG. 3 depicts assembly of plant, bearings, sensors, amplifiers,and control system in an exemplary system.

[0012]FIG. 4 shows an exemplary performance comparison between eightinterpolated H_(∞) controllers and a single lpv controller for aflexible rotor.

[0013] An exemplary high speed rotor, such as used in an energy storageflywheel, is shown in FIG. 1. The components are listed as rotor,motor/generator, shaft, magnetic bearings, static support and vacuumhousing. The example shown in FIG. 1 is an energy storage wheel incylindrical form which can rotate at high speeds. At low rotationalspeeds, the energy storage is low while at high rotational speeds, theenergy storage is very high. The energy is transformed by the attachedmotor/generator. In order to perform the desired function of energyinput, the rotor speed is made to increase by the motor/generator'sacting as a motor such that electrical energy input is converted to themechanical energy of rotation. In order to obtain energy output in theform of electrical power, the motor/generator is used as a generator andthe mechanical energy is extracted from the rotor, slowing it down.

[0014] The rotor is attached to the rotating shaft by the diagonalstruts, as shown in FIG. 1, other geometric configuration, orelectromagnetically. The shaft is supported in two radial magneticbearings, typically one at each end, and a thrust bearing (not shown).The shaft has magnetic rotor components attached to it facing thenon-rotating stator components of the magnetic bearings. The stator ofthe magnetic bearings is supported on a static support, as shown inFIG. 1. A non-rotating vacuum housing completes the typical flywheelconfiguration as shown in FIG. 1.

[0015] A magnetic bearing, shown schematically in FIG. 2, consists offour primary components: a magnetic actuator composed of an assembly ofcoil wound magnetic poles attached to a ring facing the shaft, anelectronic control system which determines the control currents in thecoils, a set of power amplifiers which produce the control currents ascommanded by the control system, and a set of sensors which determinethe shaft position in the bearing. The magnetic bearing has an automaticcontrol system, implemented electronically, to adjust the magneticbearing coil currents to control shaft vibrations due to external forcesacting on the shaft.

[0016] Some rotors, such as in certain energy storage/momentumflywheels, employ ball bearings which have a limited, finite life.Magnetic bearings have non-contact operation with the rotating shaft ofthe rotor supported in a magnetic field rather than a mechanical supportsuch as resting on the inner race of a ball bearing. This results invery long life compared to ball bearings. Ball bearings have limitedphysical life due in part, to Hertzian contact stresses. The primaryfailure point in magnetic bearings on the other hand, is the poweramplifiers or control circuits. In case of failure, these electroniccomponents can relatively easily be replaced (in ground applications orin manned space applications) or made fault tolerant, and bearing lifecan easily be extended by techniques for automatic time sharing ofremaining circuits using fault tolerant algorithms.

[0017] Magnetic bearings also do not require any lubricant, which isparticularly useful in vacuum operations such as required for successfuloperation of a high speed energy storage/momentum flywheel or otherapplications. There are a number of potential applications which canbenefit from this bearing arrangement: space applications, includingsatellite energy storage and/or momentum wheels; ground baseduninterruptable power supplies, such as for critical computer powersupplies; vehicular electric power, including automotive, bus, militaryvehicles, and trucks, and many others. Unlike magnetic bearings, ballbearings employed in high speed rotors such as energy storage/momentumspace applications require either some self contained lubricant supplyor may operate in dry contact. The low temperature normally found inspace applications severely limits possible lubricants or lubricantsupply systems. Fixed ground based applications of high speed rotorssuch as energy storage/momentum flywheels will generate very highoperational temperatures, well above the normal operating temperaturerange of conventional lubricants which severely limits possiblelubricants in this case. Dry contact lubrication ball bearing operationhas very limited expected life even at moderate speeds illustrating theadvantage of magnetic bearings for these applications.

[0018] Magnetic bearings have relatively low power loss, compared toball bearings, when operated at high speeds. In many applications suchas energy storage flywheels, the consumption of energy required tooperate the rotor in the bearings is a significant factor. At highsurface speeds, the friction loss in ball bearings becomes large unlikemagnetic bearings which generally have some rotating magnetic losses,resistance losses in the bearing coils, and amplifier/control circuitlosses which result in lower rotating loss than ball bearings.

[0019] Moreover, electronic automatic control is used in magneticbearings which support high speed rotors such as flywheel rotors. Thisautomatic control, which needs to be present to operate magneticbearings, can also be employed to assist in control of flywheel momentumor control moments. Further, the automatic control system can beemployed to significantly reduce levels of vibration. Such automaticcontrol, applied to such magnetic bearings is the subject of thisinvention.

[0020] A mathematical representation of a rotor-magnetic bearing systemis initially constructed without the control system. It can be basedupon a discretized numerical representation of the mechanical componentsof the rotor, struts, shaft, static support, vacuum housing, etc. Therepresentation is formulated from physical dimensions such as diametersand lengths and material parameters such as the elastic modulus andPoisson's ratio. One typical numerical method is the finite elementmethod. Other components in the system are magnetic and electronic.Equations modeling these components are also employed based uponphysical dimensions such as magnetic bearing diameters, pole face areas,etc.; magnetic material properties such as permeability; electricalproperties such as resistance, inductance, and capacitance; andelectronic components such as amplifier voltages, currents, andefficiency. FIG. 3 shows a typical block diagram of components of therotor-magnetic bearing system. The mathematical representation of thesystem is developed by standard engineering methods and is used toprepare the rotor-magnetic bearing system model called the plant. Thismodel does not include the controller algorithm. The method ofdeveloping the controller is described in detail later in thisapplication.

[0021] The particular variables employed in the mathematicalrepresentation may consist of physical parameters such as rotor or shaftdisplacements and/or velocities, magnetic bearing currents and/orvoltages, etc. However, such a representation may consist of manyvariables and be of very large size, unsuited to controller development.Often, the large representation is reduced in size by employing thesystem modal properties, determined by standard eigenvalue methods. Amethod of selecting only the most important modes of the rotor-magneticbearing system is used to remove less important system modes. However,the modal analysis insures that the system dynamic properties are stillproperly modeled and the controller can be properly designed. The modalmathematical representation has many less parameters to consider in thecontroller analysis.

[0022] The method of controller development starts with a mathematicalmodel. A mathematical representation of the form

{dot over (x)}=Ax+Bu

y=Cx+Du  (2)

[0023] as described below, or similar related forms, is employed in thedevelopment of the control theory applied to magnetic bearing supportedrotors and is used in this invention. The above described mathematicalrepresentation, either in terms of physical parameters or modalparameters, can be placed in the form given by Eq. (1) or similar form.In this or similar form, there are a set of vectors and matrices withdefinitions as follows. The vector x is a list or column, of rotordisplacements at selected locations along the rotor and other systemproperties such as bearing voltages or currents, known as the statevector. It is of order n meaning that there are n parameters in thestate vector. The vector u is a list or column of control inputs, suchas magnetic bearing currents or voltages, known as the control vector,with k parameters in that vector. The vector y is the list or column ofoutputs at particular locations in the system, such as displacements atthe bearings or control currents at the bearings, where particularsystem performance is desired to be obtained.

[0024] The matrix A is the matrix representing the system plant,consisting of the rotor and associated magnetic bearing physicalcomponent characteristics, which is to be controlled. The matrix B isthe matrix of system parameters relating the state vector x to thecontrol input vector U. The matrix C is the matrix relating the outputvector y to the state vector x. The matrix D is the matrix of systemparameters relating the feedthrough of the control input vector adirectly to the output vector y. The purpose of the mathematicalrepresentation is to obtain a set of equations which enables the systemdesigner to calculate a control vector u which controls the rotor andmagnetic bearing system in such a way that the output displacements,currents and other system parameters in the output vector y remain belowdesign limitations.

[0025] In high speed rotors operating over a speed range, the rotormagnetic bearing control system matrices are functions of the rotorspeed p. Thus, the mathematical representation may be written as

{dot over (x)}=A(p)x+ B(p)u

y=C(p)x+D(p)u  (2)

[0026] There are a number of controllers u which would allow themagnetic bearing system operate stabily but only one is the optimumsolution. It is desired to find the controller which is the “maximal”solution, producing the minimum vibration level and/or control currents.

[0027] A method is described in this work where a single matrix X, whichis positive definite, such that the linear matrix inequality$\begin{matrix}{\begin{bmatrix}{{{A(p)}^{T}X} + {{XA}(p)}} & {{XB}(p)} & {C(p)}^{T} \\{{B(p)}^{T}X} & {{- \gamma}\quad I} & {D(p)}^{T} \\{C(p)} & {D(p)} & {{- \gamma}\quad I}\end{bmatrix} < 0} & (3)\end{matrix}$

[0028] is satisfied for all values of the speed p in the specified speedrange. In this equation, I is the identity matrix.

[0029] The performance parameter γ in the above equation is the ratio ofthe input to output gain $\begin{matrix}{\frac{{y}_{2}}{{u}_{2}} < \gamma} & (4)\end{matrix}$

[0030] where the quadratic norm of the output vector is given by$\begin{matrix}{{y}_{2} = {\int_{0}^{\infty}{y^{T}y{t}}}} & (5)\end{matrix}$

[0031] and the quadratic norm of the input vector is given by$\begin{matrix}{{u}_{2} = {\int_{0}^{\infty}{u^{T}u{t}}}} & (6)\end{matrix}$

[0032] The performance parameter γ is the measure of the system outputto the control input effort. It is desired to keep the measured systemoutput parameters such as rotor displacement at critical points andmagnetic bearing currents as low as possible for optimum operation ofthe rotor-magnetic bearing system. Thus, a method for determining thelowest possible value for the performance parameter γ over therotor-magnetic bearing system operating speed range is particularlyvaluable.

[0033] A system representation may be achieved at minimum and maximumoperating speed. Let the minimum operating speed be denoted byp₁=p_(min) and the maximum operating speed be denoted by p₂=p_(max).These are called vertices or vertex speeds. A control vector u can bedeveloped using standard, existing control methods, such as mu synthesisor H_(∞) can be found for the minimum and maximum speeds. There aremathematical models of these two systems. The minimum speed system isgiven by the mathematical representation

{dot over (x)}=A ₁(p ₁)x+B ₁(p ₁)u

y=C ₁(p ₁)x+D ₁(p ₁)u  (7)

[0034] where the system properties A₁, B₁, C₁, D₁ are evaluated at theminimum operating speed. The system satisfies the linear matrixinequality $\begin{matrix}{\begin{bmatrix}{{{A_{1}\left( p_{1} \right)}^{T}X} + {{XA}_{1}\left( p_{1} \right)}} & {{XB}_{1}\left( p_{1} \right)} & {C_{1}\left( p_{1} \right)}^{T} \\{{B_{1}\left( p_{1} \right)}^{T}X} & {{- \gamma_{1}}\quad I} & {D_{1}\left( p_{1} \right)}^{T} \\{C_{1}\left( p_{1} \right)} & {D_{1}\left( p_{1} \right)} & {{- \gamma_{1}}\quad I}\end{bmatrix} < 0} & (8)\end{matrix}$

[0035] where the performance parameter is given by $\begin{matrix}{\frac{{y}_{2}}{{u}_{2}} < \gamma_{1}} & (9)\end{matrix}$

[0036] The maximum speed system is given by the mathematicalrepresentation

{dot over (x)}=A ₂(p ₂)x+B ₂(p ₂)u

y=C ₂(p ₂)x+D ₂(p ₂)u  (10)

[0037] where the system properties A, B, C, D are evaluated at themaximum operating speed. The system satisfies the linear matrixinequality $\begin{matrix}{\begin{bmatrix}{{{A_{2}\left( p_{2} \right)}^{T}X} + {{XA}_{2}\left( p_{2} \right)}} & {{XB}_{2}\left( p_{2} \right)} & {C_{2}\left( p_{2} \right)}^{T} \\{{B_{2}\left( p_{2} \right)}^{T}X} & {{- \gamma_{2}}\quad I} & {D_{2}\left( p_{2} \right)}^{T} \\{C_{2}\left( p_{2} \right)} & {D_{2}\left( p_{2} \right)} & {{- \gamma_{2}}\quad I}\end{bmatrix} < 0} & (11)\end{matrix}$

[0038] where the performance parameter is given by $\begin{matrix}{\frac{{y}_{2}}{{u}_{2}} < \gamma_{2}} & (12)\end{matrix}$

[0039] for the maximum speed case. A linear parameter varying controllerdesign can be achieved.

[0040] The control of the rotor-magnetic bearing system is given by themathematical representation

{dot over (x)} _(K) =A _(K)(p)x _(K) +B _(K)(p)y

u=C _(k)(p)x_(K) +D _(K)(p)y  (13)

[0041] The vector x_(K) is a list or column, of rotor displacements atselected locations along the rotor and other system properties such asbearing voltages or currents, known as the controller state vector, oforder k. The vector u is a list or column of control inputs, such asmagnetic bearing currents or voltages, known as the control vector, asdefined above. The vector y is the list or column of outputs atparticular locations in the system, such as displacements at thebearings or control currents at the bearings, where particular systemperformance is desired to be obtained, again as defined above.

[0042] The matrices A_(K), B_(K), C_(K), D_(K) are the controllermatrices. The values of the parameters obtained for these matrices areobtained by standard mu synthesis or H_(∞) methods such as given bystandard engineering control methods. At the minimum operating speed,the matrices in the mathematical representation are denoted as

{dot over (x)} _(K) =A _(K)(p ₁)x _(K) +B _(K)(p ₁)y

u ₁ =C _(K)(p ₁)x _(K) +D _(K)(p ₁)y  (14)

[0043] At the maximum operating speed, the matrices in the mathematicalrepresentation are denoted as

{dot over (x)} _(K) =A _(K)(p ₂)x _(K) +B _(K)(p ₂)y

u ₂ =C _(K)(p ₂)x _(K) +D _(K)(p ₂)y  (15)

[0044] Once a mathematical representation of the control system x_(K),u₁, u₂ has been obtained in these two forms, they can be substitutedinto the rotor-magnetic bearing system equation (x) and the closed loopsystem representation obtained. The closed loop control model has thegeneral form

{dot over (x)} _(cl) =A _(cl)(p)x _(cl) +B _(cl)(p)w

z=C _(cl)(p)x _(cl) +D _(cl)(p)w  (16)

[0045] Here the vectors w represent some system disturbances such asnoise, external forces, or similar unwanted system inputs. The closedloop matrices are given by

A _(cl) =A ₀(p)+{circumflex over (B)}(p)Ω(p)Ĉ( p)

B _(cl) =B ₀(p)+{circumflex over (B)}(p)Ω(p){circumflex over (D)} ₂₁(p)

C _(cl) =C ₀(p)+{circumflex over (D)} ₁₂(p)Ω(p)Ĉ(p)

D _(cl) D ₁₁(p)+{circumflex over (D)} ₁₂(p)Ω(p){circumflex over (D)}₂₁(p)  (17)

[0046] or similar mathematical form based upon the controllermathematical representations at the two vertex speeds. Here, thematrices on the right hand side of this expression are given by$\begin{matrix}{{{{A_{0}(p)} = \begin{bmatrix}{A(p)} & 0 \\0 & 0\end{bmatrix}},\quad {{B_{0}(p)} = \begin{bmatrix}{B_{1}(p)} \\0\end{bmatrix}}}{{{C_{0}(p)} = \begin{bmatrix}{C_{1}(p)} & 0\end{bmatrix}},\quad {{\hat{B}(p)} = \begin{bmatrix}0 & {B_{2}(p)} \\I & 0\end{bmatrix}}}{{{\hat{C}(p)} = \begin{bmatrix}0 & I \\{C_{2}(p)} & 0\end{bmatrix}},\quad {D_{12} = \begin{bmatrix}0 & {D_{12}(p)}\end{bmatrix}}}{{{\hat{D}}_{12}(p)} = \begin{bmatrix}0 \\{D_{12}(p)}\end{bmatrix}}} & (18)\end{matrix}$

[0047] The controller matrices A_(cl), B_(cl), C_(cl), D_(cl) are linearcombinations of the minimum speed design matrices A₁, B₁, C₁, D₁ and themaximum speed design matrices A₂, B₂, C₂, D₂.

[0048] A positive definite matrix X_(cl), of order (n+k)x(n+k) and alinear kth order linear parameter varying controller exists if thelinear matrix inequalities $\begin{matrix}{\begin{bmatrix}{{A_{cl1}^{T}X_{cl}} + {X_{cl}A_{cl1}}} & {X_{cl}B_{cl1}} & C_{cl1}^{T} \\{B_{cl1}^{T}X} & {{- \gamma_{1}}\quad I} & D_{cl1}^{T} \\C_{cl1} & D_{cl1} & {{- \gamma_{1}}\quad I}\end{bmatrix} < 0} & (19)\end{matrix}$

[0049] where A_(cl1) is the parameter dependent closed loop matrixevaluated at the minimum speed and $\begin{matrix}{\begin{bmatrix}{{A_{c2}^{T}X_{cl}} + {X_{cl}A_{cl2}}} & {X_{cl}B_{cl2}} & C_{cl2}^{T} \\{B_{cl2}^{T}X} & {{- \gamma_{2}}\quad I} & D_{cl2}^{T} \\C_{cl2} & D_{cl2} & {{- \gamma_{2}}\quad I}\end{bmatrix} < 0} & (20)\end{matrix}$

[0050] where A_(cl2) is the parameter dependent closed loop matrixevaluated at the maximum speed are satisfied.

[0051] The solution to this problem exists if and only if there are twosymmetric matrices R and S of size n×n satisfying the five linear matrixinequalities: $\begin{matrix}{{{{\begin{bmatrix}N_{R} & 0 \\0 & I\end{bmatrix}^{T}\begin{bmatrix}{{A_{i}R} + {RA}_{i}^{T}} & {RC}_{i}^{T} & B_{1i} \\{C_{li}R} & {{- \gamma}\quad I} & D_{11i} \\B_{li} & D_{11i} & {{- \gamma}\quad I}\end{bmatrix}}\begin{bmatrix}N_{R} & 0 \\0 & I\end{bmatrix}} < 0}{{{{where}\quad i} = 1},{2\quad {and}}}} & (21) \\{{{{\begin{bmatrix}N_{S} & 0 \\0 & I\end{bmatrix}^{T}\begin{bmatrix}{{A_{i}^{T}S} + {SA}_{i}} & {SB}_{1i} & C_{1i}^{T} \\{B_{li}^{T}S} & {{- \gamma}\quad I} & D_{11i}^{T} \\C_{li} & D_{11\quad i} & {{- \gamma}\quad I}\end{bmatrix}}\begin{bmatrix}N_{S} & 0 \\0 & I\end{bmatrix}} < 0}{{{{where}\quad i} = 1},{2\quad {and}}}} & (22) \\{\begin{bmatrix}R & I \\I & S\end{bmatrix} \geq 0} & (23)\end{matrix}$

[0052] Also, a kth order controller exists if and only if

rank(I−RS)≦k  (24)

[0053] Given acceptable matrices R and S, it is possible to constructthe closed loop matrix X_(cl) and corresponding controller matrices A₁,B₁, C₁, D₁ and A₂, B₂, C₂, D₂.

[0054] This section describes the methodology used to compute thecontroller matrices. Full rank matrices M and N of size n×k are foundfrom

MN ^(T) =I−RS  (25)

[0055] The closed loop control matrix X_(cl) is obtained from the uniquesolution of $\begin{matrix}{\begin{bmatrix}I & R \\0 & M^{T}\end{bmatrix} = {X_{cl}\begin{bmatrix}S & I \\N^{T} & 0\end{bmatrix}}} & (26)\end{matrix}$

[0056] The two vertex controllers are obtained from the above linearmatrix inequalities, Eqs. (19) and (20). The controller matrices aregiven by

A(p)=α₁ A ₁+α₂ A ₂

B(p)=α₁ B ₁+α₂ B ₂

C(p)=α₁ C ₁+α₂ C ₂

D(p)=α₁ D ₁+α₂ D ₂  (27)

[0057] where the dimensionless speed parameters α₁ and α₂ are given by$\begin{matrix}{{\alpha_{1} = \frac{p_{\max} - p}{p_{\max} - p_{\min}}}{\alpha_{2} = \frac{p - p_{\min}}{p_{\max} - p_{\min}}}} & (28)\end{matrix}$

SIMULATION EXAMPLE

[0058] A magnetic bearing controller for a flexible rotor supported inmagnetic bearings was evaluated using the linear parameter varyingmethodology of the invention. The results, expressed in terms of theperformance parameter γ, are given in FIG. 4. The rotor speed range is104 rad/sec to 832 rad/sec. The speed range is relatively large and thegyroscopic changes in the plant are quite significant as the operatingspeed changes. A comparison between eight H_(∞) controllers and onelinear parameter varying controller (LPV) is shown in FIG. 4. The eightH_(∞) controllers were designed at the specific operating speeds of 104,208, 312, 416, 520, 624, 726, and 832 rad/sec. It is easily seen in FIG.4 that these controllers produce a very low value of the performanceparameter at each design operating speed. A set of interpolatingcontrollers was developed to allow these controllers to operate betweenthe design speeds with the results given in FIG. 4. The performanceparameter γ has very low values exactly at the design points butrelatively high peaks in between those values. These high peaks may bethought of as high vibration levels or high magnetic bearing controllercurrents.

[0059] An example LPV controller was also designed for this flexiblerotor using the invention. The method employed H_(∞) controllersevaluated at the minimum and maximum speeds (vertices), 104 rad/sec and832 rad/sec. The LPV method was then used to evaluate a singlecontroller which operates over the complete speed range. The results arealso plotted in FIG. 4. It is easily seen that the LPV controller has asmooth variation of the performance parameter y over the entire speedrange from 104 to 832 rad/sec. Again, these smoothly varying values ofthe performance parameter represent low to moderate levels of vibrationand/or magnetic bearing coil currents over the entire operating range.

[0060] It is important to note that the LPV controller is a singlecontroller with better performance overall than eight interpolatedsingle speed design H_(∞) controllers. Thus, the size of the LPVcontroller is approximately eight times smaller, allowing for a muchsmaller computer capacity requirement, than the example eight H_(∞)controllers. If the computer hardware requirements and storage capacitywere relatively unlimited, a very large number of nearly optimum H_(∞)or other optimized controllers can be developed for many specific speedsin the operating range and the system performance could be very good.However, there are practical limitations of cost, weight, size and otherfactors which normally limit the controller degree of complexity andmake the LPV controller very desirable for high speed rotors operatingover speed range as compared to the standard control design methods fora time independent plant.

What is claimed is:
 1. A rotor system comprising rotor, magneticbearings, and a linear parameter varying controller for the magneticbearings.
 2. The rotor system of claim 1 wherein said controller isresponsive to linear matrix inequalities.
 3. The rotor system of claim 1wherein rotor vibration or coil current is minimized with respect tothose experienced with conventional control.
 4. The rotor system ofclaim 1 wherein the control algorithym is obtained for the magneticbearings at minimum nominal operating speed and at maximum nominaloperating speed and linear interpolation is employed between the minimumand maximum operating speed controllers through linear parameter varyingcontrol.
 5. The rotor system of claim 1 wherein the controller issubstantially stable over the range of operating speeds from nominalminimum to nominal maximum operating speed.
 6. The rotor system of claim1 wherein said controller accounts for gyroscopic effects experienced bythe rotor.
 7. The rotor system of claim 1 wherein said controllerminimizes hysteresis and eddy current losses during operation of thesystem.
 8. The rotor system of claim 1 wherein bias current is less thanone-half saturation level in the magnetic bearings.
 9. The rotor systemof claim 7 wherein said minimization is attained through application ofbias current into the control algorithm.
 10. A controller for a magneticbearing system for a rotor system, said controller being a linearparameter varying controller.
 11. The controller of claim 10 responsiveto linear matrix inequalities.
 12. The controller of claim 10 whereinrotor vibration or coil current is minimized with respect-to thoseexperienced with conventional control.
 13. The controller of claim 10wherein the control algorithm is obtained for the magnetic bearings atminimum nominal operating speed and at maximum nominal operating speedand linear interpolation is employed between the minimum and maximumoperating speed controllers through linear parameter varying control.14. The controller of claim 10 which is substantially stable over therange of operating speeds from nominal minimum to nominal maximumoperating speed of the rotor.
 15. The controller of claim 10 whichaccounts for gyroscopic effects experienced by the rotor.
 16. Thecontroller of claim 10 which minimizes hysteresis and eddy currentlosses during operation of the rotor.
 17. The controller of claim 10wherein bias current is less than one-half saturation level in themagnetic bearings.
 18. The controller of claim 17 wherein saidminimization is attained through application of bias current into thecontrol algorithm.